标签：#probabilisticmodels

A probabilistic model can describe more than the data you observe. It can also include hidden variables that capture structure you cannot observe directly. But defining that model is only the beginning. Once an observation x is available, the practical question changes: Given this x , what does the model imply about the hidden variable z ? That is the central problem of Posterior Inference . The notation is compact, but the computation is not always easy. High-dimensional latent spaces, complex posterior distributions, and interactions among hidden variables can make both the posterior itself and expectations under that posterior difficult to compute. Start with the Joint Distribution Suppose a probabilistic model contains an observed variable x and a hidden or latent variable z . The model does not treat them as unrelated quantities. Instead, it represents their probabilistic relationship through a Joint Distribution : p ( z , x ) This joint distribution describes how the observed data and the hidden variable fit together inside a single probability structure. Once x is observed, however, the question becomes conditional. We are no longer asking only how x and z relate in general. We want to know how the possible values of z are distributed given the particular observation x . That conditional distribution is the posterior. Posterior Distribution: Conditioning on Observed Data The Posterior Distribution is p ( z ∣ x ) = p ( x ) p ( z , x ) ​ The numerator p ( z , x ) contains the probabilistic relationship between the latent variable and the observation. The denominator p ( x ) normalizes those values so that the result becomes a conditional probability distribution over z . The distinction is important: The joint distribution p ( z , x ) describes the probability structure of the model. The posterior distribution p ( z ∣ x ) tells us what that structure implies about z after x has been observed. In that sense, the posterior connects the model with actual data. Pos

If you approach generative models as ""networks that create images,"" the field quickly turns into a collection of disconnected architectures. A more useful developer mental model starts one level lower: What probability structure could have produced the data, and how can we represent, learn, and infer that structure without making the computation impossible? That question connects autoregressive models, VAEs, flow-based models, GANs, and diffusion models. Their architectures look very different, but they all respond to the same underlying tension: high-dimensional data distributions are difficult to represent, learn, normalize, sample from, and reason about. Generative modeling can therefore be organized around three interacting problems: Representation: How do we represent a complex high-dimensional joint distribution? Learning: How do we make the model distribution approach the data distribution? Inference: Given an observation, how do we reason backward about hidden variables or the process that generated it? Once these three pieces are connected, the major families of deep generative models become much easier to understand. From prediction to distribution learning A discriminative model usually begins with a prediction problem. Given an input x , predict the most likely output y : f ( x ) = y ar g max ​ p ( y ∣ x ) The model focuses directly on the conditional relationship required for prediction. A generative model asks a broader question. Instead of learning only the path from x to y , it models the probability structure from which the data arises. For class-conditional modeling, for example, we can model p ( x ∣ y ) together with the prior p ( y ) and recover the posterior using Bayes' rule: p ( y ∣ x ) = p ( x ) p ( x ∣ y ) p ( y ) ​ In unsupervised generative modeling, the target becomes the data distribution itself. We assume the training samples come from some unknown distribution: x 1 ​ , x 2 ​ , … , x N ​ ∼ i.i.d. p data ​ ( x ) The model then construc